Abstract
A stochastic search method, the so-called Adaptive Subspace (AdaSub) method, is proposed for variable selection in high-dimensional linear regression models. The method aims at finding the best model with respect to a certain model selection criterion and is based on the idea of adaptively solving low-dimensional sub-problems in order to provide a solution to the original high-dimensional problem. Any of the usual $\ell_0$-type model selection criteria can be used, such as Akaike's Information Criterion (AIC), the Bayesian Information Criterion (BIC) or the Extended BIC (EBIC), with the last being particularly suitable for high-dimensional cases. The limiting properties of the new algorithm are analysed and it is shown that, under certain conditions, AdaSub converges to the best model according to the considered criterion. In a simulation study, the performance of AdaSub is investigated in comparison to alternative methods. The effectiveness of the proposed method is illustrated via various simulated datasets and a high-dimensional real data example.
Highlights
Rapid developments during the last decades in fields such as information technology or genetics have led to an increased collection of huge amounts of data
If the ordered importance property (OIP) is satisfied, Adaptive Subspace (AdaSub) converges against the optimal solution of the generally NP-hard 0-regularized optimization problem
AdaSub provides a stable thresholded model even when OIP is not guaranteed to hold. It has been demonstrated through simulated and real data examples that the performance of AdaSub is very competitive for high-dimensional variable selection in comparison to state-of-the-art methods like the Adaptive Lasso, the SCAD, Tilting or the Bayesian split-and-merge approach (SAM)
Summary
Rapid developments during the last decades in fields such as information technology or genetics have led to an increased collection of huge amounts of data. Chen and Chen (2008) argue that this model prior underlying BIC is not suitable for a high-dimensional framework where the truth is assumed to be sparse. They propose a modified version of the BIC, called the Extended Bayesian Information Criterion (EBIC), with an adjusted underlying prior on the model space: For a fixed additional parameter γ ∈ [0, 1] and a subset S ⊆ P let the prior of the corresponding model be π(S) ∝ p |S|.
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